In addition to the B36 which
@Cincydawg posted above, the US Navy's first jet carrier fighter was also a prop/jet hybrid. The
Ryan FR Fireball.
The issue was that early jets did not throttle up quickly which made them unsuitable for operations on the short runways provided by carrier flight decks. To get around this issue the Navy ordered a hybrid piston/jet fighter. It used a Wright Cyclone 9-Cylinder air-cooled radial at the front spinning a traditional propeller and a jet engine at the rear to generate higher top speed. Oddly, the jet that it used was designed to operate on AvGas so that the aircraft would only have to carry one type of fuel to power both engines.
FWIW: the Wright Cyclone at the front was essentially one half of the Wright Duplex-Cyclone 18-Cylinder radial engine in use at the time on the B29. If anyone wants to know, I can explain why Radial engines always have an odd number of cylinders per bank. Ie, there are single-bank 7 and 9 cylinder radials and double-bank 14 and 18 cylinder radials and there was even a four-bank 28-cylinder radial but note that those all have either 7 or 9 cylinders in each bank.
The Fireball was actually built and placed into service before the end of WWII but it never actually saw combat.
The aircraft was a complete failure. It was structurally far too weak for carrier operations and suffered from nose-wheel failures and wing failures throughout it's short tenure in service. Also, my take is that while in theory it was supposed to combine the best of piston planes (acceleration to get off a carrier) with the best of jet planes (top speed) I think in practice it ended up not being very good at either. Acceleration with the 1,350 HP Cyclone was certainly adequate but it can't have been remotely close to the acceleration provided for the Corsairs, Hellcats, and Bearcats by the 2,300 HP Pratt & Whitney 18-Cylinder Radial. Meanwhile, the Fireball's top speed was 404 MPH which was phenomenal by the standards of 1943 when it was ordered but was nearly matched by the Hellcats and surpassed by the Corsiars and Bearcats of 1945.
A couple years after the war ended the USN introduced the first purely jet powered carrier-based fighter which was 100 MPH faster than the hybrid Fireball.